A Polyhedral Homotopy Algorithm For Real Zeros
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We design a homotopy continuation algorithm for finding real zeros of sparse polynomial systems. Our algorithm builds on a well-known geometric deformation process, namely Viro's patchworking method. The algorithm operates entirely over the real numbers and tracks the optimal number of solution paths. In exchange, the algorithm is not universally applicable: It works for polynomial system with coefficients satisfying certain concavity conditions. More precisely, it requires the given polynomial system to be located in the unbounded components of the complement of the underlying A-discriminant amoeba. A preliminary implementation of an example from the literature suggests practical performance of the algorithm. We plan to work towards a vigorous implementation including a larger scale of examples and a software paper.
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